On planar graphs, the algorithm runs much faster:
-\thmn{Contractive Bor\o{u}vka on planar graphs}\id{planarbor}%
+\thmn{Contractive Bor\o{u}vka's algorithm on planar graphs, Cheriton and Tarjan \cite{cheriton:mst}}\id{planarbor}%
When the input graph is planar, the Contractive Bor\o{u}vka's algorithm runs in
time $\O(n)$.